Observation of Boyer-Wolf Gaussian modes

Stable laser resonators support three fundamental families of transverse modes: the Hermite, Laguerre, and Ince Gaussian modes. These modes are crucial for understanding complex resonators, beam propagation, and structured light. We experimentally observe a new family of fundamental laser modes in stable resonators: Boyer-Wolf Gaussian modes. By studying the isomorphism between laser cavities and quadratic Hamiltonians, we design a laser resonator equivalent to a quantum two-dimensional anisotropic harmonic oscillator with a 2:1 frequency ratio. The generated Boyer-Wolf Gaussian modes exhibit a parabolic structure and show remarkable agreement with our theoretical predictions. These modes are also eigenmodes of a 2:1 anisotropic gradient refractive index medium, suggesting their presence in any physical system with a 2:1 anisotropic quadratic potential. We identify a transition connecting Boyer-Wolf Gaussian modes to Weber nondiffractive parabolic beams. These new modes are foundational for structured light, and open exciting possibilities for applications in laser micromachining, particle micromanipulation, and optical communications.


REVIEWER COMMENTS
Reviewer #1 (Remarks to the Author): In this manuscript four significant findings are presented: -The experimental identification of a new class of fundamental laser modes in stable resonators: the Boyer-Wolf Gaussian modes.
-The design of a laser resonator analogous to a two-dimensional quantum anisotropic harmonic oscillator, with a frequency ratio of 2:1.
-It is demonstrated that Boyer-Wolf Gaussian modes also function as eigenmodes within a medium with an anisotropic refractive index gradient, exhibiting a 2:1 ratio between the parabolic index of refraction profiles along each axis.
-A transition is identified that connects Boyer-Wolf Gaussian modes with non-diffractive Weber parabolic beams, highlighting the fundamental role of Boyer-Wolf modes in structured light.
To the best of my knowledge, Tschernig et al. present the Boyer-Wolf Gaussian modes for the first time, obtained from the corresponding time-independent Schrödinger equation of the quantum twodimensional anisotropic harmonic oscillator with a 2:1 frequency ratio in parabolic coordinates.Subsequently, the authors elegantly design a laser resonator to generate these modes and demonstrate that experimental results align well with the theoretical predictions.They provide an analytical model to rigorously illustrate the effect and validate it using a straightforward optical setup.Furthermore, the manuscript demonstrates that the Boyer-Wolf Gaussian modes function as eigenmodes within an anisotropic gradient refractive index medium, akin to fundamental Hermite-Gauss, Laguerre-Gauss, and Ince-Gauss Modes.Lastly, the authors show that Boyer-Wolf modes are foundational elements of structured light.The Supplementary Information document contains an exhaustive analysis of the Boyer-Wolf Gaussian modes, including resonator propagation, gradient refractive index propagation, and free-space propagation of these modes.I find the demonstrations both interesting and highly relevant to the optical community.This work is likely to be a cornerstone, sparking a series of novel ideas in this specific field.Therefore, I believe the manuscript is well-suited for the Nature Communications audience.Nevertheless, I have listed below some minor suggestions for the authors to consider in a revised manuscript.
2. While the manuscript thoroughly discusses the advantages of the employed resonator design and experimental setup, it is crucial to also address potential limitations and challenges for a more comprehensive evaluation.This disadvantages, that are not mentioned in this version of the manuscript, will provide a balanced perspective for readers and contribute to a more thorough understanding of the methodology.
3. In a recent publication (https://doi.org/10.1364/OE.507030),the self-healing capability of structured beams, such as higher-order Gaussian modes, Laguerre-Gaussian beams, and Hermite-Gaussian beams, has been investigated when partially obstructed at the onset of propagation in a lens-like medium with obstacles of various shapes.It would be highly beneficial to incorporate Boyer-Wolf Gaussian modes into this context and provide an explanation regarding their potential for selfhealing and the mechanisms involved.I am confident that the authors can address this question seamlessly, offering additional insights into whether Boyer-Wolf Gaussian modes exhibit self-healing characteristics and elucidating the underlying processes.This expansion of the discussion would significantly contribute to a more comprehensive understanding of self-healing phenomena in structured beams.
Overall, I find the presentation of this manuscript to be quite good.Nevertheless, there is room for improvement, and making these minor corrections would enhance the overall quality of the manuscript, resulting in a more robust and comprehensive final product.
Reviewer #2 (Remarks to the Author): This paper is experimentally obtained a new stable laser mode in the resonator -the Boyer-Wolf Gaussian mode, and the experimental results were in good agreement with the theoretical predictions.Moreover, the pattern of Boyer-Wolf Gaussian mode can successfully transform to Weber nondiffractive parabolic beams, indicating that this new type of beam is a basic element of structured light.Although this article has these innovative elements, the content of the paper is too thin, and the discussion of the experiments and methods is not detailed enough.It does not meet the criteria of Nature Communications.I think this work would be more suitable for publication in more specialized journals, such as Opt.Express or Opt.Letters.My comments to the paper are listed below: 1, The authors only select some typical length (i.e., L=12,15,17,20 and 30 cm) of the resonator.However, the modes will be different when the cavity changes.Whether the results in Fig. 3b are obtained when the cavity length is fixed?If so, the results with other cavity lengths should be discussed.2, The article mentioned that the advantage of the cavity design in Fig. 3a is that for any pair of cylindrical lenses with _y=2_x, the resonant cavity is stable for any ≤4_x.Please give a specific explanation.3, The different sizes of mode-to-pump ratio on the Nd:YAG laser crystal will significantly affect the quality of the output beam, so, what is the optimal pump spot size in this laser device?What is the impact on the output results of this new Gaussian mode when using other pump spot sizes?4, The results in Fig. 4 are from numerical simulation or experimental observation?I think the authors should clearly explain how they obtain such results.5, I am amazing that the numerical results in Fig. 2 and the experimental results in Fig. 3b can matched with each other quite well.I think the author should provide a more detail about their experimental setup, and the size of pattern in Fig. 3b should be provided.

Reviewer #3 (Remarks to the Author):
The authors found an interesting new set of structured separable cavity modes for a cavity with a ratio of 2 to 1, based on mathematical work from 1975 by Boyer and Wolf.They implemented an anisotropic laser cavity that supports the desired modes, which agree impressively well with theory.The main contribution of this work is clearly presented.
I do have some comments on some aspects that are not central to the work, and some observations that are more related to the new contribution.I begin with two points that apply to the introductory comments and not to the main contribution.
1) There is one point that the authors mention repeatedly and that in my opiion is not strictly correct as written, or at least it is not sufficiently well justified.The statement in question is that there are only "three different fundamental families of transverse modes" of the usual rotationally-symmetric cavities: HGM, LGM and IGM.These families of modes are certainly important and have served as the basis of large amounts of work, but given the degeneracy of the system (clearly indicated by the authors), they are in principle not more fundamental than others.In my opinion, the correct statement is that these are the only three families of transverse modes that can be written as separable functions.This is a mathematical statement rather than a physical one.
From a physical point of view, one aspect that makes these three families a bit more special is that they are naturally selected when the cavity presents small amounts of simple typical aberrations like astigmatism (which leads to HG), spherical aberration (which selects LGM) or a combination of both (which leads to IG) as was shown in [1].(Notice that, interestingly, spherical aberration is a higher order aberration and cannot hence be described with the ABCD formalism, and yet a perturbative treatment does lead to the LGM or IGM.)However, other more complex aberrations might lead to other modes that, given the unitarity of the system, this will result in complete orthonormal bases.For example, a combination of a small amount of astigmatism and a small rotation (e.g. in a misaligned ring cavity) can select what is known as Hermite-Laguerre-Gauss modes, which also include HGM and LGM as special cases and are then a family of complete bases with closed-form expressions [3,4], although these expressions are not separable mathematically.Again, I do not mean to downplay the importance of the three families mentioned (which in any case are not the central contribution of this work).However, I fear that a reader that is new on the subject might understand the statements as currently written as meaning that these are the only three types of modes that can be produced byt the cavity, which is not the case.Note that there is indeed a sentence in the manuscript that reads "For this reason, spherical optical cavities only support three families of fundamental modes, the Hermite, Laguerre and Ince-Gaussian modes".
2) Similarly, while I greatly appreciate the importance of Gaussian modes, I think that the statement that "Gaussian modes are the foundation of structured light" a bit too strong.It is true that many other standard solutions, such as accelerating or propagation-invariant (sometimes referred to incorrectly as "diffraction-free") beams can be thought of as limiting cases of these modes [5].There are many aspects of structured light that do not correspond to this limit.In general, all light is structured!Again, this is surely a minor point but clarity is important, and I think one can make the case for the great importance of Gaussian modes without making such a strong claim.
Let me now make some comments related to the main contribution, which is the BWM.From what I have checked, the results seem to all be correct.Nevertheless, I would like to make a couple of suggestions that the authors could consider here and/or in subsequent work.
3) This comment echoes comment 1.From the current version, a reader might be under the impression that, once the cavity is proposed and appropriately manipulated, the desired modes emerge.However, as stressed in the article by Boyer and Wolf, this system presents "accidental degeneracy", so like the standard isotropic cavity its modes can be decomposed in an infinite number of bases.One basis that immediately comes to mind is that of the (anisotropic) HGM aligned with the directions of the anisotropy.This basis is not mentioned in the main body, but only in the Supplementary Information.These modes are likely naturally selected if the strength of the oscillator is detuned from the ideal 2:1. in the main manuscript the authors mention that they selected higher order BWM by slightly misaligning the cavity "in different ways".I would guess that this took a significant amount trial and error.I would encourage the authors to think of what perturbation of the system maps onto the operator that defines the modes and that commutes with the anisotropic harmonic oscillator Hamiltonian, in a way similar to what was done in [1] for the IGM.4) While this last issue is discussed briefly in the Supplementary Information, I think it is important to mention it more clearly on the main manuscript as well.
There is an important difference on the behavior of the BWM and the modes for the standard rotationally symmetric cavities (or GRIN waveguides), such as HGM, LGM and IGM.For the latter, the modes of the cavity or the GRIN medium are also modes in a generalized sense of free space, because free paraxial propagation can be mapped onto a isotropic 2D harmonic oscillator.This is the reason why standard Gaussian modes (be them HG, LG, IG, HLG or others) maintain their intensity profile up to an expansion and corresponding decrease in intensity.Amongst other things, this means that these modes have Fourier transforms with the same shape.The new modes do not present strictly the property of self similarity under free propagation, so the profiles plotted correspond to the profile at some plane within the cavity.As it is stated in the Supplementary Information, these modes can be expressed in terms of anisotropic HGM, which accumulate different Gouy phases.Interestingly, Fig. 14 of the Supplementary Document shows that the far-field distribution (and hence the Fourier transform) is a non-uniformly scaled version of the original mode.This property is trivial to show for the HGM in Eq. (45), and I believe that the derivation in Eqs.(43-57) shows it, but does not spell it out clearly.That is, going from the field to its Fourier transform via a Fresnel or Fractional Fourier transform, the modes return to their same shape up to a scaling, but in between they likely take different shapes.This is hard to judge from the longitudinal plots in Fig. 14, where it is difficult to tell is the relative importance of the intensity maxima changes with z.If the modes do preserve their shape up to an anisotropic stretching, this is worth mentioning, as they would be interesting as freespace modes, not only as cavity modes.Again, even a brief mention of this behavior in the main manuscript (with more detail in the Supplementary Information) would be useful.

Response to Comments made by Referees
We thank all three referees for their favorable reviews and their insightful comments.
Thank you all for your positive feedback.
We generally agree with the comments made by the referees and we revised the manuscript according to the suggestions, which have indeed improved our manuscript considerably.In the following sections, we address each of the comments individually and explain the changes made in the manuscript and in the Supplementary Information in response to each comment.

Point by Point Response to the Comments of the Referees
We now address each comment individually.For each point, we first present the referee's original comment, followed by our response.To facilitate this process, we have assigned numbers to every individual point raised by each referee, organized in the order of their appearance in the report of each referee.

Reviewer #1 (Remarks to the Author):
In this manuscript four significant findings are presented: -The experimental identification of a new class of fundamental laser modes in stable resonators: the Boyer-Wolf Gaussian modes.
-The design of a laser resonator analogous to a two-dimensional quantum anisotropic harmonic oscillator, with a frequency ratio of 2:1.
-It is demonstrated that Boyer-Wolf Gaussian modes also function as eigenmodes within a medium with an anisotropic refractive index gradient, exhibiting a 2:1 ratio between the parabolic index of refraction profiles along each axis.
-A transition is identified that connects Boyer-Wolf Gaussian modes with non-diffractive Weber parabolic beams, highlighting the fundamental role of Boyer-Wolf modes in structured light.
To the best of my knowledge, Tschernig et al. present the Boyer-Wolf Gaussian modes for the first time, obtained from the corresponding time-independent Schrödinger equation of the quantum two-dimensional anisotropic harmonic oscillator with a 2:1 frequency ratio in parabolic coordinates.Subsequently, the authors elegantly design a laser resonator to generate these modes and demonstrate that experimental results align well with the theoretical predictions.They provide an analytical model to rigorously illustrate the effect and validate it using a straightforward optical setup.Furthermore, the manuscript demonstrates that the Boyer-Wolf Gaussian modes function as eigenmodes within an anisotropic gradient refractive index medium, akin to fundamental Hermite-Gauss, Laguerre-Gauss, and Ince-Gauss Modes.Lastly, the authors show that Boyer-Wolf modes are foundational elements of structured light.The

Supplementary Information document contains an exhaustive analysis of the Boyer-Wolf
Gaussian modes, including resonator propagation, gradient refractive index propagation, and free-space propagation of these modes.

I find the demonstrations both interesting and highly relevant to the optical community.
This work is likely to be a cornerstone, sparking a series of novel ideas in this specific field.
Therefore, I believe the manuscript is well-suited for the Nature Communications audience.

Our response:
We thank the referee for the positive reaction on our manuscript.We are pleased that the referee found our work both interesting, highly relevant to the optical community, and well-suited for publication in Nature Communications.As described below, we have carefully implemented the referee's suggestions, which have significantly enhanced the quality of our work.
Nevertheless, I have listed below some minor suggestions for the authors to consider in a revised manuscript.

Our response:
We thank the referee for carefully reviewing our manuscript and for finding this typo.We have corrected the sign in the equation accordingly.

2.
While the manuscript thoroughly discusses the advantages of the employed resonator design and experimental setup, it is crucial to also address potential limitations and challenges for a more comprehensive evaluation.This disadvantages, that are not mentioned in this version of the manuscript, will provide a balanced perspective for readers and contribute to a more thorough understanding of the methodology.

Our response:
We thank the referee for bringing this matter to our attention.Similar to a spherical resonator, the primary limitation here is the presence of high-order aberrations.As we describe in the manuscript, the theoretical description of any paraxial ABCD resonator only considers quadratic terms in position and momentum.Consequently, this implies that higher-order aberrations are missing in the conventional ABCD ray formulation.The overall effect in our work, as in any other spherical laser resonator, is that the misalignments that we introduce to generate higher order modes need to remain within the paraxial regime, i.e., small misalignment angles.We have now included a paragraph to the manuscript where we discuss the limitation of paraxial resonator design and experimental setups.Additionally, we highlight approaches for studying this system beyond the paraxial regime.

Paragraph:
"The primary limitation of this formalism is that the theoretical description of ABCD systems only considers "quadratic optics" and does not include higher-order aberrations, such as mirror imperfections and non-paraxial corrections.Here, and generally, the paraxial approximation is entirely adequate because deviations from this approximation are usually small when the resonator mode waist  is significantly larger than the wavelength .Therefore, for our paraxial resonator, these corrections only become important for misalignments outside the paraxial regime, for very high-order modes, and for resonator configurations with very small waist, i.e.  ∼ 2 .A complete theory of resonator aberrations for nonparaxial beam propagation and optical elements beyondquadratic mirrors and lenses can be found in Ref. [48]." 3. In a recent publication (https://doi.org/10.1364/OE.507030),the self-healing capability of structured beams, such as higher-order Gaussian modes, Laguerre-Gaussian beams, and Hermite-Gaussian beams, has been investigated when partially obstructed at the onset of propagation in a lens-like medium with obstacles of various shapes.It would be highly beneficial to incorporate Boyer-Wolf Gaussian modes into this context and provide an explanation regarding their potential for self-healing and the mechanisms involved.I am confident that the authors can address this question seamlessly, offering additional insights into whether Boyer-Wolf Gaussian modes exhibit self-healing characteristics and elucidating the underlying processes.This expansion of the discussion would significantly contribute to a more comprehensive understanding of self-healing phenomena in structured beams.

Our response:
We thank the referee for bringing to our attention this interesting recent work.Following this reference, we have conducted a similar analysis of the self-healing properties of the high-order Boyer-Wolf Gaussian modes propagation in a 2:1 GRIN medium.As the referee suggested, we found that similar to the higher-order Laguerre, Hermite, or Ince-Gaussian beams, the Boyer-Wolf modes also exhibit self-healing properties.The mechanism is the same as the one presented in the reference above.The main difference is that since the BW modes are eigenmodes of a 2:1 GRIN medium, as opposed to a 1:1 GRIN medium, the self-healing process is dictated by the axis with the larger period of oscillations.
We have added a sentence to the manuscript addressing the self-healing properties of the BW based on this new reference.More importantly, we have included an entirely new section in the Supplementary Material (Section 7.1), where we conduct a more thorough analysis of the self-healing properties of the BW modes following the suggested reference.Furthermore, we illustrate this phenomenon with two new supplementary figures (Supp.Fig. 19 and 20).Overall, I find the presentation of this manuscript to be quite good.Nevertheless, there is room for improvement, and making these minor corrections would enhance the overall quality of the manuscript, resulting in a more robust and comprehensive final product.

Our response:
We are glad that the referee found our manuscript quite good.Indeed, implementing the referee's suggestion has enhanced the quality of our manuscript.

Reviewer #2 (Remarks to the Author):
This paper is experimentally obtained a new stable laser mode in the resonator -the Boyer-Wolf Gaussian mode, and the experimental results were in good agreement with the theoretical predictions.Moreover, the pattern of Boyer-Wolf Gaussian mode can successfully transform to Weber nondiffractive parabolic beams, indicating that this new type of beam is a basic element of structured light.Although this article has these innovative elements, the content of the paper is too thin, and the discussion of the experiments and methods is not detailed enough.It does not meet the criteria of Nature Communications.I think this work would be more suitable for publication in more specialized journals, such as Opt.Express or Opt.Letters.

Our response:
We are confused by the comments of the referee.We believe the referee probably did not realize that there was a supplementary section to the paper.As the reviewer #1 said "the Sup. Figure 20 | Partial and fully self-healing of higher order BWG modes.We show several x−y-slices of the propagation shown in Fig. (1).At z = L z /4 and z = 3L z /4 we observe (almost) complete healing of the beam profile after encountering the circular block.At other distances we observe only partial healing or, in the extreme case, the recurrence of the initial circular block.Sup. Figure 19 | Propagation of partially blocked high-order BWG mode.We show the x − zslice of the propagation of the n = 44, l = 0 BWG mode in a 2:1 GRIN medium with a = 1.1272 1/m and λ = 1064 nm.The blocked region causes the emergence of shadows that propagate along sinusoidal paths and reconstitute the initial blocked circle at z = L z = 2.786 m.